a) An axisymmetric convex 101-gon is given. Prove that its axis of symmetry passes through one of its vertices.
b) What can be said about the case of a decagon?
In each cell of a \(25 \times 25\) square table, one of the numbers 1, 2, 3, ..., 25 is written. In cells, that are symmetric relative to the main diagonal, equal numbers are written. There are no two equal numbers in any row and in any column. Prove that the numbers on the main diagonal are pairwise distinct.
A coin is tossed three times. How many different sequences of heads and tails can you get?
Each cell of a \(2 \times 2\) square can be painted either black or white. How many different patterns can be obtained?
How many ways can Rob fill in one card in the “Sport Forecast” lottery? (In this lottery, you need to predict the outcomes of thirteen sports matches. The result of each match is the victory of one of the teams or a draw, and the scores do not play a role).
In a football team (made up of 11 people), a captain and his deputy need to be chosen. How many ways can this be done?
How many six-digit numbers exist, the numbers of which are either all odd or all even?
There are five books on a shelf. In how many ways can the books be arranged in a stack. (Stacks may consist of any number of books)?
\(N\) young men and \(N\) young ladies gathered on the dance floor. How many ways can they split into pairs in order to participate in the next dance?
The Russian Chess Championship is made up of one round. How many games are played if 18 chess players participate?